Welcome to CalculaGames!

Very, Very Educated Guesses — An Intuition for Taylor Series (and π)

By CalculaGames • Updated April 2026

You’re listening to someone rattle off hundreds of digits of π.

At first, your brain starts to lose track—you begin to think of π as some kind of endless serial number.

But then you remember: each new digit is another level of precision.
Tenths. Hundredths. Thousandths. Millionths…

And at some point, a question creeps in:

How do we know π this precisely?

It can’t be from measuring circles.

Even if you took the largest imaginable circle (the observable universe), and measured its circumference and diameter with the smallest meaningful unit (the Planck length) you would only get something in the order of a few dozen digits of π.

But we know trillions.

These are not just different amounts of accuracy.
They are way different kinds of accuracy.

So you pull out your phone and look it up.

You might find something like this:

π = 4 × (1 − 1/3 + 1/5 − 1/7 + 1/9 − 1/11 + …)

An infinite alternating sum of fractions.

That might satisfy you.

Or it might make things worse.

If you’re in the latter camp, this article is for you.

The short answer is: calculus.

But even if you’ve never taken calculus, there’s a surprisingly intuitive way to understand what’s going on.

A Car at an Intersection

Imagine this statement:

“At 9:15, there is a car at the intersection of Bear Valley and Main.”

Do you know where the car will be in the future?

Not at all.

It could be parked. It could speed off. It could reverse.

You know where it is, but you know nothing about where it’s going.

Now add more information:

“At 9:15, there is a car at the intersection of Bear Valley and Main.
At exactly 9:15, the car is moving 60 km/h down Main Street.”

Now you can predict where it will be in the near future.

But only roughly.

You still don’t know if the driver is speeding up or slowing down.

Add another layer:

“At 9:15, there is a car at the intersection of Bear Valley and Main.
At exactly 9:15, the car is moving 60 km/h down Main Street.
At exactly 9:15, the gas pedal is depressed 2 cm.”

Now you know the car is accelerating.

Your prediction improves.

But there’s still uncertainty. Are they pressing further? Easing off?

One more layer:

“At 9:15, there is a car at the intersection of Bear Valley and Main.
At exactly 9:15, the car is moving 60 km/h down Main Street.
The gas pedal is depressed 2 cm.
And the driver is easing off the pedal at 1.5 mm per second.”

Now your prediction improves again.

Notice what’s happening.

You’re not seeing, learning, or measuring the car’s future path.

You’re reconstructing it from how the car behaves at a single moment in time.

Each new piece of information lets you see farther into the future.

What This Has to Do With Math

These layers have a mathematical name: derivatives.

  • Velocity is the rate of change of position
  • Acceleration is the rate of change of velocity
  • And so on…

Each layer describes how the previous layer is changing.

Now imagine a graph instead of a car.

  • The car’s position → the function’s value
  • Velocity → the slope
  • Acceleration → how the slope bends
  • Higher layers → finer behavior

There’s a special kind of function called a polynomial:

x, x², x³, and so on.

Polynomials are important because each term directly encodes one layer of behavior:

  • constant → position
  • x → slope
  • x² → curvature
  • x³ → change in curvature

They are like perfectly organized containers for this information.

Here’s the key idea:

If you know enough about how a function behaves at one point, you can build a polynomial that mimics it.

This is called a Taylor series.

It takes all those layers, position, slope, curvature, and beyond, and stacks them into a sum.

Each new term adds another layer of accuracy.

Where π Enters the Picture

So here’s the unexpected connection:

π lives inside certain functions.

Functions like sine and its inverse (arcsin) naturally produce values involving π.

That means we can do something clever:

  • Take a function that contains π
  • Rebuild it using a Taylor series
  • Evaluate that series at the right point

And out comes π.

That alternating sum you saw earlier?

1 − 1/3 + 1/5 − 1/7 + …

That’s not random.

It comes from building one of these Taylor series and evaluating it at a specific point.

Each fraction is like one more “layer” of behavior being added in:

position + velocity + acceleration + …

So How Do We Know π So Precisely?

Not by measuring circles.

But by doing something much more powerful.

We find a function that contains π.

Then we use its behavior at a single point, its value, its slope, its curvature, and so on, to rebuild it piece by piece.

And when we evaluate that rebuilt function…

out comes π.

Every additional term in the series is like adding another layer to the car snapshot:

position → velocity → acceleration → …

Each one lets us “see” a little farther.

And just like predicting the car’s path into the future, we are predicting π further and further into its decimal expansion.

So we don’t know π because we measured it perfectly.

We know π because we found a process that can keep refining it forever.

You don’t know where Rachel’s house is.

But you know a car is heading there.

And not just that. You know everything about the car right now:

  • where it is
  • how fast it’s going
  • how it’s accelerating
  • how that acceleration is changing
  • how that rate of change is changing

With enough of that information, you don’t need a map.

You can predict where the car will end up.

That’s what we’re doing with π.

We’re not measuring it.

We’re following something that leads to it, and we're using layers of behavior to get there.